Oscillating slender ships at forward speed

OSCILLATING SLENDER SHIPS

AT FORWARD SPEED

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP

AAN DE TECHNISCHE HOGESCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS IR H. i. DE WElS

HOOGLERAAR IN DE AFDELING DER MUNBOIJWKUNDE VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP VRUDAG17 DECEMBER DES MIDDAGS TE4 UUR DOOR

WILLEM JOOSEN

WISKUNDIG INGENIEUR GEBOREN TE BERGEN OP ZOOM

Dit proefschrift is goedgekeurd door de promotor PROF. DR R TIIviiMAN,

CONTENTS

Suuvt&aY 8

CHAPTER I. INTRODUCTION . 9

General introductiOn. . . 9

The thin ship theory . . . 11

The slender ship theory 12

The effect of forward speed V 14

CIA.PrER II. FORMULATION OF THE PROBLEM 15

Co-ordinate system and definitions 15

Linearization of the equations and conditions 17

Outline of the approach 19

The existence and uniqueness of the solution 22

CHAPTER 1H. THE STEADY STATE CASE FOR HIGH FROUDE NUMBER . 25

The linearized velocity potential . 25

The linearization of the free surface condition 28

I The wave resistance 30

CHAPTER IV. THE CASE OF HIGH FROUDE NUMBER AND LOW FREQUENCY . 33

The velocity potential 33

The asymptOtic expansion. 34

The limiting cases for L = O and

= 0

38

The critical value y = 0.25 40

CHAPTER V. THE CASE OF LOW FROUDE NUMBER AND HIGH FREQUENCY 46

Introduction 46

The steady state problem 46

I The unsteady case 48

The case of Froude number zero 55

CHAPTER VI. THE ADDED MASS- AND DAMPING COEFFICIENT FOR THE

HEAV-ING MOTION 57

The hydrodynamic forces 57

The low frequency approximation 58

The high frequency approximation 59

CHAPTER VII. NUMERICAL RESULTS 60

APPENDIX 66

RereIcEs

69

ABSTRACT

A potential heory is developed for slender ships trávelliñg and oscillatthg at

the free water surface For the steady state the linearized problem is solved and

a non hnear correction term is derived Ship forms with minimum wave re

sistanôe are obiained. Due to the correction term the curve of sectional areas is not symmetrical, this in contrast with previous results. For the unsteady problem two different approximate theones are derived One for the case of high Froude number and lw frequencies and another for low Froude number and high frequencies. The effect of forward speed on added mass and damping coeffi-cients is considered. Finally the numerical results are compared With ex-perimental data.

CHAPTER I

INTRODUCTION

I GENERAL INTRODUCTION

The object of the present dissertation is to derive under certainconditions, a systematic and self-consistent theory for the motion of a partially immeEsed

slender body at the surface of a fluid.

The problem of the motion of a solid in an unbounded liquid is in general difficult to solve The reasonis that the Navier Stokes equations of motion, by which the disturbance in the fluid is mathematically characterized, are very

intractable. A wáy to simplify the problem is to assume that no viscous effeóts are present. The effect of the fluid friction, which is most important close to the

body, especially for the calculation of the viscous drag, can then be treated

separately with the aid of the boundary layer theoty.

A further resriction that can be made is to assume that the medium is incompressible Of course this assumption is only admissable if the disturbance velocities are small in regard to the velocity of sound in the fluid.

In the following the attention will only be paid to the study of those aspects

of the problem for which the viscosity and compressibility effects are negligible.

Täking into account the Kelvin theorem on the conservation of circulation, it is obvious that a motion, which is irrotational 'at one moment of time, will remain irrotational. The theorem is still valid if an external conservative force

is present, such as gravity.

Under these circumstances the motion of the fluid is characterized by the existence ofa single valued velocity potential function It can be shown [i] that in an uñbounded fluid is uniquely determinate Save for a ôonstant if it

fülfils the following conditións:

a. The Laplace equation or what is equivalent, the equation of continuity. b The normal derivative of cJ at any pòint of the surface of thè body is equal

to the normal velócity of the surface á.t that point (Neumann problem).

c; All derivatives of 'F vanish at an infinite distance in any direction from the body, due to theconditiôn f thiite kinetic energy at infinity

By application if Green's theorem on the function 'F an integral equation for F is obtained. The proof for the existeñce and uniqueness of the solution can be found in the textbooks [1].

Another formulation of the Neumann, problem is to express 'F in surface integrals over a source distribution oñ the body. The unknown distribution is

determined by an integral equàtioñ which is derived from the boundary

condi-tion on the body. From this Fredhoim equacondi-tion a unique solucondi-tion can also be

obtained [1].

Recently HEss and SMITH have solved the problem of an arbitrary body by

a numerical method using source distributions. The only restriction in their

In the past several approximate solutions had been obtained by assuming certain geometrical restrictions. One of the best known concepts is applied in the so called lifting surface theories. It is assumed there that the body is thin,

only slightly cambered and situated in a uniform flow. If the camber and thick-ness are characterized by a small parameter the velocity potential and boundary

conditions can be expanded in ternis of this small parameter. It can be proved that for the derivation of the first order term in the series of the potential it is consistent to put source and dipole distributions on the mean plane of the surface. After calculation of the source and dipole strength the pressure

distri-butions along the surface can be obtained. However, no resultant force on the body is found. In order to bring the mathematical model in closer agreement With the actual physical situation the condition that the potential is single valued is dropped. In this way the concept of a circulation flow around the

surface enters the theory and leads to a lifting force. The strength of the

circula-tion is determined by the condicircula-tion of smooth flow at the trailing edge of the

surface.

Another shape, on which further simplifying assumptions can be imposed is the. slender body. Here the dimensions in the two lateral directions with regard

to the main flOw are small in regard to the third. The same procedure, as used

in the thin body theory, can be applied in this case. However, a difficulty arises.

In thin body theory it was possible to satisfy the boundary condition on a flat plate. In the slender body theory the body shrinks down to a line in the limit case. In this case it is difficult to apply boundary conditions on a line for the three dimensiónal differential equation. This difficulty can be avoided by representing the body either, by a line distribution of sources, if the cross sec-tion shape is circular, or by a distribusec-tion of multipoles for an arbitrary shape. The integrals for the potential are written in a co-ordinate system, where the co-ordinates transverse to the body are stretched in the ratio of the reciprocal of the slenderness parameter. In these new co-ordinates the body remains finite if the limit is taken for the slenderness parameter tending to zero. After expansion of the potential function the boundary condition can be applied and taking into account the first order term only, it appears that the source strength is proportional to the axial derivative of the curve of sectional areas. A necessary condition in this approach is the fact that the body is sharply pointed.

The fact which makes the present work much more complicated in comparison with the theories mentioned above, is the presence of a free surface. The subject

of water waves engaged the attention of many mathematicians and physicists during the last century. A review of their results is presented in an excellent

survey by WEHAUSEN and LAITONB [3]. Only those aspects will be given here which. are ofinterest to the problem discusse4.

In addition to the conditions stated in the foregOing the velocity potential must satisfy another boundary condition at the free surface. This non linear condition is originated from two others by eliminating the elevation of the surface. The. first condition is a kinematic one and prescribes that there is no lo

transfer of fluid particles across the surface. As second condition the pressure is assumed to be constant along the surface. In order to keep the problem linear it is necessary that the slope of the waves is small.

Fundamental potential functions can be constructed, which have the saine

singular behaviour as travelling or pulsating sources and dipoles in an

unbound-ed munbound-edium and which satisfy in addition the linearizunbound-ed free surface condition. In order to determine these functions uniqúely the usual boundedness.

condi-tions at infinity are not sufficient. It is 'therefore necessary to impose at infinity the so-called radiation or Sommerfeld condition, which excludes' the existence of incoming waves generated at infinity A more general way to ensure the uni-queness of the solution is given by STOKER [4]. He formulated the problem as an initial value problem by assuming the medium originally at rest. When the time tends to infinity the solution tends to the desired steady state solution.

Then the only conditions needed are those of boundedness.

Ifa solid body of finite dimensions is moving at thç free surface, the problems of existence and uniqueness Of the solution for the potential are in general not yet solved. A discussion on this subject will be postponed till chapter two.In a

long series of papers Havelock has treated problems. related with the motion of special bodies at the free surface by representing them by source or dipole

distributions.

The same restrictions for the body shape as in the case of an unbounded medium can be made for the free surface problem. The next two sections deal with the essential features of the thin body and the slender body theory in relation to the free surface effect.

2. THE THIN SHIP THEORY

The study of thethin ship model was initiated by MIcHELL's paperin 1898 [5]. He derived a formula for the wave resistance of a ship moving at finite speed in

smooth water. The linearized free surface condition was used and therefore it was necessary to assume the ship to be thin 'in order to ensure that the gene-rated waves are small. Michell assumed a source-distribution at the center

plane of the ship. At this plane the linearized böundary condition was satisfied

in order to obtain the source strength. After derivation of the formula for the

velocity potential the wave resistance could be obtained from energy flux con-siderations at infinity

Especially the last years it was sometimes felt that Michell's assumptions were adhoc approximations and that 'in addition to the condition of small beam/length ratio other conditions were imposed in relation to the other linear dimensions in the problem i.e.. the draft and the ratio between the square of the ship speed and the acceleration of gravity (MARu0 and VossERs [6], [7]). Recently WEHAUSEN [8], showed, however, that Michell's formula can be

ob-tained from the original th±ce-dimensional formulation by a rigorous asymp

totic expansion with respect to the small beam/length ratio only.

for a comparison of experiment and theory. A striking agreement was seldom

found, however, the main difficulty being the experimental determination of the

wave resistance. In most cases this is obtained by subtracting the estimated

viscous resistance from the measured total resistance. Following this procedure

it is moreover assumed that no interaction effects. occur. In order to check Michell's theory. it seems to be more reasonable to compare its resiilts with

those obtained from a more exact theory. There are two approximating

condi-tions in Michell's theory: the linearization of the boundary conditioñ on the ship's hull and the linearization of the free surface condition. The only way to construct an improved consistent theory is to drop both assumptions, which leads to -a very difficult and intractable problem. Another possibility is to maintain only. the linearized free surface condition. Then a two dimensional integral equation for the velocity potential can be set up by Green's theorem. This approach, however, has not yet been made. For a further discussion of

these linearization problems is referred to OGILvIE's paper [9].

The first attempt to a rational approach of the unsteady thin ship problem

was undertaken by PILERs and SToKER [10]. They treated a ship in sinusoidal

head waves, with an amplitude of the same order of magnitude as the beam. Peters and Stoker started from the exact non linear problem and assumed an expansion of all variables in perturbation series in powers of the small

para-meter, the beam/length ratio. These expansions are all substituted in thè. various

conditions and equations and the terms are all arranged according to powers of the beam/length ratio. It appears that for the lowest order of this parameter in the resulting differential equations of motion for heave and pitch, the added mass and damping terms are not present. Therefore the solution predicts an undamped resonance. The spring constant is the hydrostatic restoring force and the disturbing force is the socalled 'Froude Krylov' force. The latter is the force obtained by integrating the pressure in the incident waves over the hull.

In order to avoid this difficulty NEwM.&N assumed more than one small

para-meter in the problem [11]. He introduced two other parapara-meters; one related

to the order of magnitude of the incident waves and another related to the order

of magnitude of- the unsteady motions. It is then possible to express the last parameter in the former two parameters. This relation is different for the two

cases, viz. at rèsonance and at non-resonance.. At and near resonance the added

mass and the damping forces are dominating. Unfortunately, however, the

resulting expressions are rather complex.

3. Ti-ia SLENDER SHIP THEORY

The slender body theory is based on the assumption that both beam and

draft are small compared with the length of the ship. The procedure to construct

a slender ship theory is essentially the same as for a slender body in an un-bounded medium. The derivation of the theory, however, is much more

corn-licated due to the free surface effect..

by VOSSERS [7]. Although his general approach has proved to be very useful the

elaboration of the theory is not correct at some places The general concept is the formulation of an integral equatiòn for the velocity potential by means of Green's theorem. This integral equation was simplified by expansion with respect to the slenderness parameter. He concéntrated his attention on the problems of the steady advancing slender ship (high Froude number) and the oscillating ship at zero forward speed (high frequency). After his work was

published several papers by others appeared.

A different method was applied by URSELL [12] and TUCK [13]. Both authors used the technique of innerand outer expansions for bodies of revolution, a well

known method in the theory of viscous flows at low Reynolds numbers. Tuck treated the problem of the steady moving ship and Ursell solved the problem

for an oscillating ship at low and high frequencies. The latter derived a second order theory in the slenderness parameter.

JOOSEN [14], [15], formulated both problems for an arbitrary shape of the body with the aid of wave source distributions and obtained the same results

as Tuck and Ursell as far as the first order term in the expansions is concerned.

Moreover he omitted the condition that the body is sharply pointed. It appears then that the series expansion is ñot uniformly convergent anymore in the

neighbourhood of the endpoints.

Following Vosss method, NEWMAN [16], [17], derived the solution fOr the

unsteady motion of an arbitrary pointed body in oblique waves. He restricted himself to low frequencies or in other words to that range of waves, where the

wave length is of the same order of magnitude as the ship length. In the lowest

order theàry for yaw and sway there are three kinds of forces occurring in the

equations of motion, the inertia forces, the Froude Krylov force and the mOtion

induced forces In the latter there is no free surface effect Moreover no

inter-action effects between the sections occur In fact the result is the aheady known two-dimenional strip theory. In the first order theory for heave and pitch motions the same difficulties arise as in the thin ship theory. Due to the fact that no added mass and damping terms are present the theory breaks down at resonance. A bounded resonance, results, however, from the secOnd order

equations. A disadvantage of this result is that in case of high resonant peaks it seems' difficult to understand this phenomenon as a second order effect.

Nevertheless it is' still possible that the theory gives good results, especially if the theory can be extended to the case of forward speed, because only there the reso-nant peak is in the range of frequencies, where the theory is 'assumed' to bevalid. Apart from these usual slender body'theories' another result is obtained if the

motion of a slender body oscillating at high frequencies is'considered, more

precisely for the case, where the frequency parameter has the order of magnitude

of the reciprocal of the slenderness parameter. Joosen proved that with this assumption for the case of heave and pitch Grim's two dimensional strip theory is resulting GiUM [18] developed his theory some years ago from physical

reasoning. The disadvantagé of the strip theory is 'that the three-dimensional

to extend the theory in order to take forward speed effects into account. In order to obtain sorne insight into the interaction effects Grim proposed to use a strip theory for calculating the singularities representing the ship and then to determine the flow velocities with three-dimensional potential functions. In principle it is of course possible to work out this theory for the forward speed case as well. However, the objection against this approach is that it is not rational and consistent.

After the foregoing discussions and conclusions it is evident that the next logical step in the development of the theory is to include the forward speed êffect. But then this theory must be worked out in the same systematic and

logical way as fOr zero forward speed. Before starting the derivation of such a theory a few general remarks will be made on the forward speed effect.

4. Ti-is EFFECT OF FORWARD SPEED

The calculation of ship motions in regular waves has been discussed in a number of papers (KORvIN KROUKOVSKY [19], VASSILOPOULOS [20]). In these papers the, two-dimensional strip theory was used to calculate the added mass and damping coefficients (GRIM, TAsM [21]).

The effect of forward speed on the hydrodynamic forces was considered and

dynamic cross coupling terms were included in the equations of motion for

heave and pitch. Korvih Kroukovsky introduced in his theory terms depending on speed which are, however, mathematically of the second order of magnitude. Vassilopoulos obtained with this theory a good agreement between theoretical and experimental values for the motions. GERRITSMA [22], [23] showed

ex-perimentally that the effect of speed on damping, added mass and exciting forces is rather small. The effect on the dynamic cross coupling coefficients, however, appears to be important. Recently he measured the added mass and damping distribution along the length of the ship, during forced oscillation tests with a niodel moving at varióus forward speeds. He found a very small effect of speed on added mass distribution and on the total damping force.

These values were in striking agreement with the theoretical values, calculated by Grim's strip theory for zero forward speed. The damping distribution varies

largely with speed. Gerritsma added to the damping distribution, obtained by the strip theory, the second order correction term of Korvn Kroukovsky and

found a good aeement with the experimental data. It seems, however,

dangerous to draw conclúsions from this fact without knowing the compan

which inclûdes all secoid

wonder whether there is a first order ten to explain the discrepancy.

The object of the following is to look into the details of.this possibility by expanding the complete three-dimensional equations and conditions in series of the small parameter in a systematic and self-consistent way. Two different problems will be considered; the case, where the frequency parameter is small and the Froude number is large and the case, where at a high frequency the

CHAPTER II

FORMULATION OF THE PROBLEM

1. CO-ORDINATE SYSTEM AND DEFINITIONS

In the co-ordinate system used in the following the (x1, y1)-plane coincides with, the mean plane of the free surface and the(y1, z1)-plane is through the

midship section. The origin moves with the ship speed Vinto the saine direction as the ship and the z1-axis is taken positive in upward direction.

The immersed part of the ship at time 1, is given by an equation of the type H(x1, y, z1, t,) = O

In eqùilibrium position fór t = O the hull surfaçe is assumed to have the

form

Yi = f1 (xi, z1) gii Yi

The length of the ship is L, the beam B and the draft T. A cross section contour is indicated by C(x1) The bow and stern contour are denoted by Pb and P3 respectively The bow and stern have the shape of sharp wedges In the next pages BB' and SS' are designated respectively as the bow region and

the stern region.

Only heaving and pitching motions of the ship, harmonic in time with angular speed c, will be considered. The displaçements and rotations of the ship are supposed to be small with respect to the ship dimensions and are defined by:

L

ic1

= a-0e

and 4 =

where is a dimensionless small constaht.

Although only the pure harmonical motion is considered here, it is clear that

super-position principle is accepted. The result can be built up by taking the sum of

the various fre4uency components For further reference see [9]

Since it is assumed that the medium is inviscid, irrotational and incompres-sible,

the flow velocities can be characterized by a potential function

J*(x1, Yi, z1, t) in the following way:

u (x1, y1, z1, t) = grad 'I (x1, Yi, Z1, t)

with

*(v, y,

?i, t) _ -: Vx1 + 0(x1, Yi, z1, t)

where I0(x1, y, z, t) is the disturbance potential and detex4nines the

distur-bance velocities due to the presence of the ship in the uniform flow.

The ship is supposed to be slender i.e. the beam/length and draft/length ratio are characterized by a small parameter .

In order.t facilitate the mathematical elaboration of the fonnulae it appears

t be UsefUl to intrOduce the follo'wing dimensionless quantities:

L L

'L

Yi = z1 = f1(x1,z1)sgny1 =

O gL B 2T

t = -

0I(X1, j'' z, t) = 2 - (, ij,.

O):

= -

d =

-Co 2co L B

2v2 co2L o2B coV

-gL 2g 2g g

The dimensionless co-ordinate normal to the surface of the body is denoted

by 'i. In the new co-ordinate system the transverse and longitudinal dimensions of the ship are of the saine order of magnitude.

2. LINEARIZATION OF THE EQUATIONS AND CONDITIONS

In this section the equations and conditions for the potential function

(, m

) will be derived It is evident that cI must satisfy the Laplace

equation in the halfspace Outside the body

The. behaviour of (I) at infinity will be discussed later. Theñ the only remaining

conditions are the free surface condition and boundary condition on the body Since no fluid particles can be transported through the surface of the body the condition at Ji -. O is:

Hg - VH ±

H, + (J)

H,, +

H = O

or in dimensionless form:

H0 - yH +

H +

JJ, +.

H =. O

If the elevation of the free surface is denoted by h (x1, Yi) the kinematic

con-L . 2

ditioñ at z1 = h is

. ₂ . . --- Vh

h - (1,, h,

L

The dynamic condition at z1. = h_ follows from Bernoulli s law:

- VD0, ± gh

_±

1

(2 + cD20,, + 2) +

O

22

p

where P Is the pressure, which is assumed to be constant at the free surface

These two conditions take the dimensionless form:.

-h

r2 1 1

±

=

_at

h

LheY, + £Ih1 +

= O

-After eliMination of h the result is

+

±

+

_(-

+

+ 2j +

+

+

±

+

+

+

+

+ L00 ±

ycb0 + = O (2.2).

Iñ order to linearize the pföble the expressions must be expanded with :rspect to a small parameter.

There are two independent small parameters. The first one, the slenderness

parameter ,is related to the geometry of the ship and the other one is a, which

is related to the amplitude of the oscillations. It seems difficult to decide apriori

in what manner h and.(I) are depending on and a. Therefòre it will be asumed here, as usual that I and 1, are of lower order than h, 'D21,1 and ct2.

Con-sequently and 'I 1 can be neglected.

Further it is supposed that a Taylor series expansion of '1 in the neigh-bourhood of O exists. Then it appears to be consistent to satisfy the linear-ized free surface condition at = O assuming that h is at most of order 2 In

LO6 +

- 2y

+

= O at

= 0

(2.3) After solviñg the problem it is of course necessary to check the result, since it

is possible that it is in contradiction with the assumptions.

The slenderbòdy theory in an unbounded medium (WAIW [24]) suggests the possibility of this contradiction. The: results show that the expression for and (I), contains ternis of order In and terms of order unity, whereas is of order unity. Therefore the term must be included in the formula for the pressure on the body. For the linearization of the boundary cOndition on the body the equation of the hull surface will be expanded with respect to and a

tO

H(1, , , O)

f(, )} -

-

1)fe + O(a2, a2)

In order to satisfy condition (2.1) the potentia.l is splitted Up into a part depending on time and into a part, independent of time.

, ,

O) = yI(,

, + '1)2(1, h, , O) (2.4)

With the assumkion that 0(1) <L <0 (e_1) the relation (2.1) becomes:

f, +

+

+ Q(2I), a) = O

on

= f(,

₎

(

0j)fe+

ï

(o

«i, (o

-+ D2 -+ fr,

+ O (,

a2L,

c32)

= O on

= f(,

From the last condition it follows that 2must have the form

iO

2 (1' i,

O) = «)2 (, h,

) e

For the steady and unsteady part of the problem the following twQ sts of equations and conditions are now obtained:

For the steady part:

Li=o

Vi + f2,

+ O(e) =

,

+

= _f

For the unsteady part:

L2 +

+ 2k421 ± «)

= O (2.6)

Vi

+f!2

_{+ O( ) =}

+ (t2f, = -

L(O

oi)f, +

+ y

{0

_-

(o

-

+

(o

-

oi)

fr,

Moreover the radiation condition must be satisfied by both potentials at infinity (chapter I, 2 and chapter II, 4).

3. OIJTLÌNE OF THÉ APPROACH

The velocity potential Will be fäthulated in terms of a source singularity

diStribútion on the Ship's hull

,, :)

=f df F(,

i = i stea4y case

I C

i 2 unsteady case (2.7) where Gi is a Grèen's fünction satisfying Laplace's equation, the free surface condition and the radiation condition at infinity F( ) is the strength of a singularity distribution to be determined from the boundary condition on te body The existence of a unique solution for F(, ) will be discussed in the next

section.

F is a source distribution multiplied by a factor (1 + f2 + 2f2)1/2 since for

a surface element dS can be written

dS = (1 [2 + c2f2)1/2 dd It appears to be useful to introduce another definition

(2.8)

with Po

=fdf

F(,

)2 + 2 (

f)2 ± )2

+

)2 ± D2 ± £2(

+

)2} d

The form of Gjôan be fouñd e.g. in [9]:

G2(1, i,

; , ,

=

01 oo + )q + i(E)qcos O

r

Ç qe

cos{1 f)q sin 6)

d

j 0q2 cos 0 ± 2yq cos 0 +

L q

dq +

O O

+ )q + i(, cos 0

2 r2

r

qe

cos{s(1 f)q sin O)

--id0i

_,tj

_{j 3q2 cos2 O + 2q cos O + L}

dq+

q 0 M1

c(+ )qi(1)qcos 0

2 ç2

r

qe

cos{(i1 J)q sin O)

--ido,

dq 7rJ

j0q2cos20-2yqcos.6+q

o M1'

withcosO1=-41r

G1(1,1,1;

=

[G2(j,i,i;

.

If the roots in he denominators are denoted by. q1, q2,

M1 and M2 are defined by

H

q2

i

2y cosO + ii/41 cos O-1

q1 = q1 = _20cos2O

i + 2y cos O -

V'i+ 4y cosO

q3

2f3 cos2 O

1+2-ycosO+\/l+4ycosO

q4

. 20cos2O

The aim is the derivation of the lowest order term of p and P2 in the series expansion with respect to In other words the iimiting values of Pi and P2

____q3 ______q4M2

I

i

--

2y cos O ï\/4y cos O i

q2=q2

_2Ç30cos20

i

2y cos 0 V'i

4y cos O

q2 23 cos2 O q1 = I 2y cös O + V'i 4y cos O 23 cos2 O cos 01 <

±

4y (2.10) q3, q4 the contours cos 01 > i

will be derived when. tends to zero. One may expect, however, that this

theory will give a good approximation for the case of small finite as well Itis quite easy to obtan the first order terms for the potential p0 Changing

the order of integration in the first integral the result after oneintegration with

respect tO becomes:

1d

) d

j

J

. )2 ± Ê2 f)2 +

I

i .

() d

+fin

2

I1_dfF) d ±

_f

Iii 2IE

fF

t)4

±fF(

) li

--

±

+ V(1

+ Ê2(

+ 2

} d

+fF(

i {

+..

d± Ó(1n)

(2.11) During the derivation it was assumed that the bow and the stern region are óf order . If these regions are Of order Unity the result can be written in the forni:

) In Ê

_{--f)2 + (}

)2 d +

+fsgn

_(i

) In 2j Id For Po obtaiñed Po =

2fP(1)

{ln \/(1_f)2

In

i/(.1_f)2

± (±} d +

±

2fF(,) [i g,1 + V(1_)2 + 2 (f)2 + 2(_} +

lñ gj +

+ 2(D2

±

_±

)2}]

d +

+

2fF()[lni_

E ± /(_)2 + 2(1f)2 +

)2} ±

- in {

+

V(1 )2 2(

_f)2 J

2( +

d (2.1 2

where the last two integrals can be neglected, if the body is sharply pointed. Before starting the derivation of the lowest order terms in p and P2 it is necessary to introduce some statements as to the order of magmtude of a, Ç3 and L with respect to As was mentioned earlier three cases will be considered

1 a = o

Ç3= 0(1) L = o (2.13)

2 a = a

f30 := 0(1) L = O(I)

cc « 0(1)

(2.14)

3 a = s2

f3o f3i

L = B'

f3 = 0(1), B = 0(1) oc <0(1)

(2.15)

where oc ìs a constant factor.

The first problem is the steady state case for high Froude numbers. [ second case is elated to the problem of low FroiÎde number and high fre-qUencies and oe may expect that it corresponds with a ship moving in head waves of small length. The third case deals with the problem of high Froude

number and low or modetãte frequencies and it seems to be a good

approxima-tion for a ship moving in following waves with moderate length. The first problem will be. treated in chapter III the second problem in chapter IV and

the third problem in chapter V In the next section the existence and uniqueness of the solution will be discussed as far as the lowest order terinsare concerned.

4. ON THE EXISTENCE AND UNIQUENESS oF THE SOLUTION

There are several possibilities to reduce the Numann problem to an integral

equation.

The potential has to satisfy: the Laplace euation.

the condition that the normal derivative at the hull is equal to the normal

velocity compónent.

3 the condition that the derivatives in any direction are vanishing at infinity

the free-surface condition.

the radiation condition at infinity.

The constant value, which can always be added to the solution of a Neumann problem is taken as zero. This causes no loss of generality Since only the induced velocities are physiôally of interest and these are not affected by the

cOnstant part of the velocity potentiâl.

One possibility is to start with the application of Green's theprem

47c(I)Ç1, Yi. z1)

=

J',f(I)(x.y,.z) G(x1, y, z1; x, y, z) dS ± s + s, + s, + s,

The Green's function G satisfies Laplace's equation and behaves in the neigh-bourhood of (xj, y1, z1) as:

G-ffn(xï j', ) G(x, Yi' Zi , y, z)dS

s±s+s+s,

i

\/(xi_x)2 + (Yi y)2 + (z_z)2

Since should satisfy the radiation çon4itio the same con ition must be imposed on G. At infinity G tends to zero of sufficient order to make the integrals over S2 and S3 vanishing,

For a steady forward motion the free surface condition reads

g'D, + VcD,

O at O

If G also satisfies

gG ± V2G

O at z1 O

the intégial over S can be integrated with respect to x1 and the result iñ dimensionless co-ordinates becomes:

4i

=

fJ'Gdd

_ff«öa4

_±

-G) d

L

Another possibility, which is often used in potential theory [1] leads to an integral equation which is sometimes more convenient for computation and

it will therefore be used in the present work. A solutiòn of a Neumañn problem

can be represented in terms of a source distribution on the hull surface In tius

case the sources will be Kelvin sources, which satisfy the free surface condition

and the radiation condition.

,

= ffF(E

) Ç(i,

; ,

) dd

where F (, ) is to be determined from condition 2. Assuming that a soh.ition

for the functiòn F exists is expanded close to the body in a series of terms in .

It will be proved in the next chapter that the first order term has the following form:

b(1,

,

= - 2f

F(1, ) {ln V(1 _/)2 + (

+

C

± in

_{J)2 ±}

₊

} d ±

)

where T ( is a function of1 only. The unknown source strength is determined

by the integral equation

(,

=

- 2rr

F(1,

) +

\/ 1+f21

in

_f)2

₊

)2 + l

v'(1 -J) + (

+

} d

C

where is a known function on the hull. This equation is the same as for the two dimensionál problem in an unbounded medium. It is well known that this

problem has a unique solution Thus it can be concluded that as far as the first order term is concerned, F exists and is uniquely determined The bow and stern region however must be excluded from this consideration, if the ship is not sharply pointed.

with

z(L ± )q + ic(m f)q sin O + i(L - cos O

2

t2

te

dq - Re I dO I

7C TJ

j Ç30qcos2O 1

2 L

Re means that the real part of the integral is to be. taken and L represents the

contour in the q-plane

It appears to be useful to change the form of G slightly. Using the expression:

i

)2_{± 2(i11}

_J)2

*

)2

=_RefdJf'e

1 dq

1 s(+q + ic(--f)q sinO + i(5)q cosO

- o G1 becomes:

i

)2 +

_f')2+ 2(

i ')2 c(, + )q + (m f)q in O + i(- - cos O

20

f2

fe qcos2O

---ReidOl

dq

t

j

j Çq cos2 O - i

L

Putting this expressIon in (3.1) the first or4er term in the series expansion can easily be obtained The first term of G1 leads to a result already denved, see

(2 11) On the second term in G1 an mtegration with respect to can be applied Omitting higher order terms in the result.has the form:

CHAPTER

ifi

THE STEADY CASE FOR HIGH FROUDE NUMBER

1. T LINEARIZED VELOCITY POTENTIAL

It is.asstthied that the veloöity parameter o is of. order unity and that the

ship is shafply pòinted. According to (2.8) the velocity potential is given by:

7h i) Po(i 7h

) ± pi(' 1i'

where p0 follows from (2.12). The second term becomes:

Pi('

, )

=fdf

FOE, )G(1, 7h ; , , )d (3.1)

1

C

+

with

leads for g1 to

2(3 dp - cosh

gi(j, ) =

Re jJç30p

j e

ch

The integration contour L can be changed by closing it to the upper side if

> O and to the lowr side if - < O. The:separation of the -intërvl

is máde in order to ensure the convergence of the integrals.

)g1 (i

i - )q coo O

2(3e e cosO

:g1(1,

) = -

ReifdO/p

-Introduction òf the new variables

qcos2 O = p cos O 00 idp

__)pcoshr

g1(, ) = - Re

iJ'

_-

_i/e

dr

>0

)

4fF(1,)

+ (

+

d +

+2fln2I

2/in

2I - jd

JF(.

)d +

i coshr

00 00

00i(j)

2 i dp )P cosh COSbt

=Rei{-13ofjí3p+1fe dT

.=-.ife

dT or

4o fK(p1

=

322 ±

dp

4o rK(pII)

- TI

2oP + dp

where Y0 is the Bessel function of the second kind, and K0 the modified Bessel

function of the second kind The integrals can be reduced fuflher by the rela-tion: j)

o)

_12P2±l

dp=

O

fI1I

Y01-\ F-10 I \ Po 27 Here Ho is the Struve fuñction. Finaiiy the form Qf the velocity potential becomes: 1li

= -

2fF(1,

)(1n

\/(

f)2 +

)2 + C

+ in rV(1 f) + (

+

)2)

d +

+

2f

S) sgn (

- )

in 2j1

_{Id ±}

+

sgn

-,

y0

(IiI))

d +

47fS) y(Ii -I)

_{d (3.2)} with

S()

=fF(.

d

The first two terms can be considered as the potential aSsociated with thó motion of the double body in an unbounded medium The last two ternis

<o

- > o

represent the free surface effect. The source strength F is determined by (2.5)

(I)

±

=

Since the free surface contribution depends on only, this leads to the integral equation

11m 2

(F(1

)

f +

-

d

( ) C

J

(i - f )2J

)2

= fi(i'

i)

C+

C is the contour reflected with respect to the , plane.

47vF(1,

+

2f F(i,

₎

d = f1, )

c+C

where the principal value, of the integral is to be taken. An integration is carried

out over the area of the double body. After changing the order of intègration

the result becomes:

fF(1

)d -

fF(

d1 = A,()

(3.3)

where A () is the área of a cross section. The inner integral is the potential of a doublet distnbution along the contour with strength umty and equals zero

Outside the cOntour. Therefore

fF(1 )d

A,(E)

c+ and

S()

=

I

A)

47

2. Tii LINEARIZATION OF THE FREE SURFACE C0NI)ITION

The a priori assumption made in section (I, 2) will be discussed here. In deriving the free surface condition it was assumed that the quadratic terms in the Berlioulli pressure formula are negligible or more in detail the term con-taming 12 From the expression (3 2) it follows that the velocity potential

(I) coilsists of a term of òrder ln and another term of order unity. The derivative

in 1-direction is at leäst of order c in . Thederivativein ï1-direction, however, is of order unity 2 .rF( )'

hf

d

Im

J

1'

+

2 (

_f)2 ± ( +

C (3.4)

This expression is the same as for the unbounded medium prOblem, where it is

known that the linearized form for the pressure on the slender body ïs not

satisfactory.

At a first glance this could lead to the conclusion that the use of the linêarizèd free surface condition is not consistent and that the condition shoild read:

+

- o (-

±

+ Ö(2 in

) = O

If, however,

or1 are of order

' the order of magnitude of and is not unity but . Taking the correct limit for tending to zero one must there-fore distinguish between the near field behaviour and the far field behaviour of

the conditions:

near field far field

In the theory developed here, the latter expression for the whole field is used. Since in the near field the first term dominates over the second, the cOrrect first order terni of I) is obtained. It can. be concluded_therefore that the use of the linearized free surface conditionleath to a mathematithily consistent theory.

Althoùgh not being consistent with the theory we shall try to obtain some

indication of the non4inear free surface effect.

For the velocity potential is taken the function

-

= O

the free surfáce condition becomes in the whole field

11

+ r0111 = O

and the function can be Obtained as before. The only problem s to satisfy (3.5). The boundary condition on the body gives on the line of intersection of the body with the free surface

ct11(1, _,

l) ± g()

in the near field

m ) far field

The free surface condition becomes:

+

+

(g1

2)

-

= o

.(I)

+

= O

By determining g () in such a way that

near field far field

= f(1, o)

In order to obtain a rather rough approximatiön for g it is assumed that the last expression is valid in the whole region close to the body Consequently

can be neglécted. Then the result becomes

g=f,o)

This result leads to the conclusion that a non-linear correctiòn term can be

added to cIa, which is only approximately valid and has the magnitude

g(1)'

o) d

It seems somewhat unrealistic that the correction term is of order unity and

does nOt depend On the Froude number. If the exact free surface condition could

be satisfied it can be expected that the resulting term is of higher order and depends on the speed. Due. to the approximation used in the procedure above these characteristics of g have disappeared. Nevertheless it is felt that g will

give at least an indicatiön of the non-liñear effect.

3. Tiii WAVE RESISTANCE

The wave resistance of a slender ship can be calculated by integrating the

pressure over the surface of the ship:

=

2L2fd1f

P(1, ,

1)f1, 1)d

where the pressure P follows from Bernoulli's law which is for the steady case

P = 2pV2

+

+

After integrating once with respect to E the dimensionless form for the wave

resistance becomes: R

= pV2L2 -

Jdff1

g1(i' , )

d1

The only contribution to the wave resistance arises from that part of '1 which

depends on the Froude number:

R = c4fA

1)[fA)

sgn H0(R1 I)

0(Ri_

or

R=-j d

(36)

By using the integral representation for the Y0 function this expression can be

written as

R=

4f {r2(r + 52(T)) dT

where

r(r)

=fÁ)

cos (- cosh

r) d

5(r) Sifl (- cosh r) d

lt is clear that this fúnction is always positive. It can be shOwn that the non-symmetrical ship form increases the wave resistance without change of the

displacement Therefore the form for minimum resistance should be symmetric

According to the non-linear behaviour of the velocity potential however a

term can be added to the formula for the wave resistance. This correctión term is

=

o) d (3.7)

LR is zero if the curve of sectional areas is symmetricàl wjth respect to = O R can have a negative value for ship forms with a non symmetrical curve of

sectional areas In that case the wave resistance is decreased by an amount of It is therefore to be expected that the optimal ship is not symmetrical any more, if the ¿R is taken into account The optimal form will be determined by a method based on the variational calcules The conditions under which the

wave resistance is minimized are

A(l)=A(l)=O

A(1) = A(l)

O

o) k(), a known function

If 3 h () is a Sm.11 variation on the function A () with h (1) = h t-!) = h (1) = h (-1) = O, then the condition for nununwn resistance at a fixed displace-mentis

-

(R+LR+XD)=.O

(3.9)

where A () is replaced by A () + f3h (a). After a partial integration (3.9) can bewritteñ as

I i

_fh(1)

_fA()Yo('

d -

f'1(1,o)

_?]

d1 .= o

This relation must hóld for any arbitrary h () and therefore

2

k() - =0

1

or

fA(Y(1

d

fk()d

+ a2 ±. b

+ c

(3.10)

with the presöribed conditions (3.8)

...

= 0

=0

(3.11)

2D

1

From the relations (3.10) and (3.11) the function  () can é caïculated. The cuÑe of sectional areas follows then from

CHAPTER IV

THE CASE OF HIGH FROUDE NUMBER AND

LOW FREQUENCY

1. ThE VELOCÍTY POTENTI L

In this chapter the derivation of the theory is restricted, to the linearized

problem. The Froude number and frequency parameter are both supposed to be

of order unity This case seems to be a good approximation for the problem of

'r

a ship moving in waves with moderate wave length or more precisely waves -

/

o2L Yt_

which are characterized by the condition. that the parameter -i--- is of order unity. Here c is the frequenôy of encounter

/ \2

c2L icL/

ii

rLt

L,i_c0sV13o) =0(1)

= direction of the waves.with respect to the positivea-axis. X = wave length

The steady part öf the velocity potential is already derived in the foregoing

chapter:

pi(i'

i,

) = -

4f F(1;

) ln 's./(

_f)2 + (

)2 d Lj

±

2fd

fF()4[sgn

_{( - ) in 2I - I+ H ('}

sg

(i

The unsteady part of the potential P2 is again of the fòrm

P2(1, , )

=JdfF(E, )G2(1,

, ; , ,

) d.

(4.2).

where G2 for this case is written as:

G2-

.

+;+

)2 + _{f)2 4} 2 (

with

.01 oo_ + )q + i(1 - COS O

-

rd

ÇAn(q, ø)e cos{c(v1 f)q sin ø}

n

-J

q2 cos2 O ± 2yq cos O ± L q

q ±

o o

)

+

c(, ± )q ± i()q coo O

-

r

4n(q, O)e cos{r( f)q sin O}

7CJ

J

oq2 cos2 O + 2 yq os O + - qdq +

r ÇBn(q, O)e cös(1 -.f)q e}

7Ç!

J0qcos2O_2(qcoso_q

A4

=

0q2cos2 O ± 2yq cos O

B4 Ç30q2cos2 O - 2yq cos O

2. T ASYMPTOTIC EXPANSION

The determination of the first order term ofP2 is possible by application of

the same procedurè as. used in chapter III. The term containingG4 is integrated

with respect to . It is assumed that the ship is sharply pointed and therefore

fF(1, )d ==fF(_1, Od

=

The first order term in P2 can easily be obtained by putting = O in the formulae containing G, since the inteals remain all convergent The result becomes:

P2 = -

4JF(E,

)l

Where O, co i(, _{)q cos O} 2 r An(q, O)e dq G

= j

dOj

0qs2 O ± 2yqcos 0+ q

+

o o

i(11 i)q coo O

2 r20 Am(q, O)e 'dq

-j

d j oq2cos2 O _{+ 2yqcos O + q}

O M1

34

- f)2

4: _{+ 132}

+

±

2fsgn( ) ln 2I - 13d1 ±

+fìft,

13G3& ±fG4 fF(,

(4.3)

n = 3, 4

(4.4)

COSO

2 r2

IB(q, 0)e

__JdeJ

dq

30q2cos20-2yqcos 0 + Ejq

o M With. A3, = B3 =

A4 = - i(Çq cos O + 2y) B4 = i«30q cos O

-After ¿hangih the integratión contOur in themanner as was done in chapter III the result for the Gteen's function is:

If Ô ., IqCOSO 21 ç

r4(ïq, 0)e

=

J

d0j

q2 còs2 O + Z1Y COS O - IqCOSO 21 f

f B( iq, O)e

_{d +}

+

_J

'10J ç0q2 2 _{O2i-q COSO + L +}

.

cos O _dO

+

4iJA(qi O)e

4y cos

+

(

c6s O _dO (4.5) O)e cos O o

and if > O:

__I(iIqcosO 21 f2

ÇAn( iq, 0)e

G

= j

dOj _{0q2cos2 :0 2iyq cos 0 +}

o o

__jIqcos

O

2i (

rBn«, 0)e

7J

J

Ç0q2cOs2O=-2iyqcos 0 + o o

-4if4n(q

E)qrcos O )ç

+ L

dO

V'I 4y cos 0

i (

cos O _dO

+

±

4if

B(q3 O)e

V'i +

ï coS 0 o dq + L + ¡q -

dq+

¡q 35

where

R3 = j.

R4 = 2i1 ± p sgn (- )

After integration with respect to O:

00

ÇR(p) e

Gfl')o=21

dp

J V'{

í3oP + 2iyp sgn ))2 +p With the notations

= (L

1P2)2

('hï - l)P2

m2 = 4yP(L - p)sgn (

the final form become:

=

i/fRn(p){(/mi2

_{+ m22 + m1)1/2 +}

i sgn m2(/n21 + m22 mi)h12)e V dp

(4.7) V'm12+m22

Ïhi function has a 'logrithmic singularity for

=

. The integrals in (4.5) and (4.6) orinating from the residues, can be transformed intO a form, which is more convement for computation Introducing the new vanable

20qj cosO . V h, i, c, d

= and denptng:

V V H3 C3 L3. D3 =

i(E)q4cos O

O)e

r

.i ( -

cos O

+

Aa2, O)e. O 00 4

IIp

= -f R(p) e

_dP/{

dO

V'i +4y

cós O dO v'4y cos O

The first two integrals of(4.5)and (4.6) can be reduced to a more simple fOrm

by the substitution V

qcos O=p

dO

- PoP2 +

2iyp sgn (j

+

O (4.6) S2 O ± p2

the result is

for - <O:

ib,.

=

8ihf

H(T) e

for - i> O

H4 =

_i(_2)

C4 = D4 =

i(r

-

2ï)

L4 = -

i(r

+

c= i +2y--V'1 +4y

d= I +2y±V1+4r

h=1-21\/1-4y

if1<

¡ = i 2'y + Vi

-h=1=2y

ify>

00 id

-

--

Sidf

Da(r) e dr

=

8icf

Cn('r) e

- 2) 4c2'r2

dr

+ 811fL() e2°

V(lr + 2y)4

4/2

+

d'r

- 2y) - 4dr2

For the last integral of (4.5) and (4.6) the new variable i - 2y cos O cos 6 isintroduced. Denoting (4.8) 37 dr V(h'r + 2T)4

- 4h

8=4y1 N3=L

the following expression is obtained

Ify>

Ify<

(49)

From the formulae it becomes clear that isacriticai valué. The behav-iour of the velocit' potential in the neighbourhood of this Value will be

dis-cusséd in (IV, 4). First the limiting values for L = O and - O will be dérived and compared with previous results Although L and1 are both supposed to

be of order unity it will appear that the limiting value for one of the parameters tending to zero leads to a result, which is also obtained if the parameter is taken

equal to zero beforestarting the asymptotic approximatiòn.

3. T LIMITING CASES FOR = O AN]) = O

The limiting valueof G for L = O can easily be obtained by putting L O

and y = O in the expressions for G. The result becomes:

G3(1) G3(2) := G33) = = O O G4(') = 2Ç30 sgn

_T

IV'(3O2P2

₊

dp

2 sgn ( -

)f

e d'r

-- (I-2r)(t)

G(3)=-4e

= *sg

-for 00 00

_j;(ti_)

.>O:

G4(2)=4fdr+4fdr

47Y(1

_I)

Ri - ti {V6( _{+ 1) (I}

_{)iIsgn (,)}}

fN() e

+ 2) I)

The final result is therefore: G3 = O

G4 = H0(

ío

sgi (ai

)} 0

(1zr

I)

and this is in agreement with (3.2), the free surface effect cf the velocity potential

for the steady advancing slender ship, without the non-linear correction term 1 _o Ô for is obtained G) = G4(2 = G3(3) = G4(3) = O G3(1)

₌

2Lf;,7

p -

- -

I)} o

L(;)

- 1 <°

G(2) =

4iLf

dr r : C,1 ,

G2) =

4iLf,i

d' + 8ilim

/OOS1fl O d The last integral is zero, except for

- ,

O. It is clear, however, that this term does nOt produce a contribution t the velocity potential and therefore

it is correct to takê

G) = 4i

o

=2 rn

L{Jo(Lji

) + i HO(L[F,J - I)}

and for G is obtained

G3 =

L{2iJo(Lji

-I)

HO(LIJ

YO(i

I)}

G4 O

From (4.4) follows the expression for the potential P2

P2(1, ,

l)

-

4f

F(i, 1) ln

j)2( +)2d+

C

±

7rLf{2i

_{Ho(LIiE) - Yo(LiI)} d}

(4.10)

The final form for G3 was earlier obtained see [12], [14],. [16] It áan easily be derived froth the.original form (4.2). Putting f3 = O in (4.3) G3 can be written as:

re

J(p4./(

)2 + (m

_d

G3=7cEj

PEL

where L represent the contour

Taking the limit for tending to zero the result is:

rJ0(i'II)

G3=rrLJ

dp

-

7CL{2i Jo(LIj -

I) -

Ho(LIi -

I) -

YO(Li I)}

4. ThcRmcALvAuEy=0.25

The fOrmulae (4.8) show clearly that if y approaches to 015 thé velocity potential has a singular behaviour The term in the Gren's function G which becomes singular in the neighbourhood of y = 0.25 is G(2):

With

? = 2\/1 -- 4y,

i. = 2V'4y 1

-Gl6iHe

L.

and y tending to 0.25 from the lower side:

ç'

dr

G(2)16iHe

I.

1 V(r+ 1)2±4r{1(lA)}

r

- .dr

J

\/{(+ i)2±4} {(l +À) i

and ify approaches to 0.25 from the upper side:

i I

dr G(2)

l6iHe

G1(2) 161 H e

f

dT

{(r--

1)2+4r}{(T_l)2+ (L2} For A -0 or - O the final result becomes:

G

G(2) 2 '\[2 i H1 e

In Il

-

4yJ

Therefore it can be concluded that the damping and added mass coefficients

tend to infinity when y tends to 0.25.

This phenomenon has been examined by various authors HASKIND [25 J,

Bit&iw [26], HANAOKA [27], and HAVELOCK [28] for a thin body. Havelock

mentions that the remarkable behaviour of the wave motion at the critical point could be illustrated in considering the two-dimensional problem of a

line source at the surface of the water. In the following the velocity potential of such a line source will be derived.

A source of strength unity is situated in the origin. The source pulsates hannonically with frequency - and the incoming flow has the velOcity V in the

positive x-direction. By application of the result, derived in the appendix, the

velocity potential, owing to the source can be written as

yp+ixp yp-ixp

fe

dp

fe

_dpìe

p(x,

=

j

2 2

j

₂ J

M,p2+2yp+--pM,p2-2yp±--p

The poles of the integrands are:

1+2yV'l+4yg

_{l+2yH-V'l+4yg}

2 V2

k4=

2

-

V2

After changing the integration contour in a way as done in chapter III for the

integrals there the result bécomes:

for x <0:

00 iyq - IxIq 0° -iyq - xlq

fe

dq

fe

p(x, y)e

=

J

2 _Co2

+

j

2 _Co2

+

o - - q2 --- 2iyq + - - ¡q

o - - q2 -

2zyq +

-

+ iq

-

1>

°

l-2vV"l-4

i

-

1-2y±iV'4y1 g

k=

V21 2 k1=p1= 2

l2y_i4y_lgY>

k2.

_{2 V2}

for x> O:

iL

p(x,y)e =

2ti

(y+ïx)kj R2 = e

if y <.

- 2it

(

i-R2=

/ e

ify>+

v4y1

At lafge distances from the source the velocity potential behaves as represented in the sôheme below

x<O

P = W2 + W3 + W4 w3 ± w4

2,ti (y ix)k 2ici (y

-+:

e

-

e

+R1

Vl+4y

\/l+4y

2ti

(y + ix)k

Rl=e

ify<.

2 (y+ix)a R1 =

-

-

e if y > + 00

-

_klq Ç e

'J

V o 00 (yqxjq __

f

el

w3=

Vi + 4y

ky - i(k,x+ ù) e dq

l' ---q2 +2iyq+--

+ R2 where: 2*i ky + i(kx W1 e

-2,ri ky + i(kx ot)

e

v/i 4y

dq

2

+

43 2rri k1y - i(k + c,)

w4=

e

Vi + 4

For better understanding of the wave phenomena it seems easier to consider

the waves in a co ordinate system winch moves with the velocity V in the

direction of the undisturbed flow

X = X + Vt Then the result becomes:

2ri ky + «Ic - _g k1V2 V

w1=Vf4e

1V(Y+) CgI.114

2rd ky + i(k - g /

k V\

V w2= - e

2IY+I

c2-Vl-4y

V\

_{g J}

g

l+\/1-4y

27ci k,, - i(k, + (),t)

g/

kV2\

V w3= e

(i)3=Hy -=I

c3=

V1+4y

V\

g /

g

Vi+4y-1

2rri k4y -

-

_g k4 V2 V

w4=Vl+4e

w1 and w3 are outgoing waves travelling in the direction óf ± bo

and - oo

respectively. w2 and w4 are incoming waves from - oo The condition for progressing waves in a liquid of infinite depth is satisfied:

k1g = The phase velocity of the waves follows from

(I).

C. = -k

and the group velocity becomes cgj cj. Fôr <0.25 the four different waves exist. If y> 0.25 then w and w2 disappear. The amplitude of these two waves becomes infinite at the critical point y = 0.25 and therefore also the kinetic energy in the waves per unit of length tends to an infinite value. This becomes clear by considering the fact that the group velocity of these two wavés equals V and consequently the energy cannot be carried away towards infinity.

Apart from the above derivation the essential features of the wave system can be explained by a rather simple consideration Let ci be the frequency of

an oscillating source in a uniform flow with a velocity V Then the disturbances

onginated by the source have the same frequency o in a co-ordinate system fixed to the source We focus our attention to solutions of the problem, which have the character of progressing waves. Two different kinds àf solutions are

For waves with a phase velocity, which is independent of the frequency, such,

as accoustic waves, only one frequency IL is possible:

IL = co(1

Gravity waves however have to saisfy the dispersion relation, which reads in

this co-ordinate system

p.2 = kg and consequently

g

IL

The frequency IL can be obtained by substituting (4.14) in (4.13)

V

This equation leads to two solutions

1V1-4<V/g

IL

-2 V/g

(4.13)

(4.14)

waves moving in the positive x-direction with a velocity potential

=cos(rnxct)

(4.11)

and waves moving in the negative x-direction

= cos (mx + cat) (4.12)

where m is the wave number, which is related to the wave length A as

27c

A

Let us now consider these waves in a co-ordinate system moving in positive x-direction at speed V, which means that the velocity of the medium is zero in this system. The frequency of the waves is nOw denoted by p., the wave tnumber by k and thê phase velocity by c. The relation between these quantities is

.

-

IL = ck

44)1'd

For waves of the type (4.11) the relation between and k becomes

ci)=(cV)k

and

1+V'1-4coV/g

2V/g

In the co-Ordinate system, which is fixed to the source, these two waves have the same frequency co but different wave lengthgA. For small values of V the

behaviour of co1 is:

co1 = ++

g In the limiting case for V equal to zero:

= co and co2 does not exist.

This is e'ident since the two co-ordinate systems are identical then. For small

values of co the behaviour of co2 becomes

co2

and thus for co = O is obtained:

co1 = O and co2 =

-V

This is in agreement with the well known fact that a non-oscillating source

travelling at speed V creates a wave with frequency

₃

For waves of the type (4.12) the relation between co and is

i V\

co= I1---lL

\

cl

which leads to and

V'i + 4coV/g i

.L=co3

-2 V/g

v'i +.4coV/g+ 1

= co4

-2 V/g coV g

If - =

it follows that i = co1

₌

_{2 =} and therefòre: c = 2V

Consequently the group velocity of these waves becomes equal to V, which is exactly the speed of the source. lt is clear therefore that no waves can exist

CHAPTER V

THE CASE OF LOW FROUDE NUMBER

AND HIGH FREQUENCY

1. INTRODUCTION

In this chapter the free surface contribution in the velocity potential will be derived for another speed and frequency range than in the forgoing chapter. Froude number is defined as

₌

and the frequency parameter as

=

where both Ç and B areof order unity The supposed order of magnitude of the Froude number seems to be in good agreement with the range of most practical interest The values of the frequency parameter correspond to small wave lengths in regard to the ship length It is expected that the main result for this case will be the two dimensional strip theory which is already used in practice Its use ha always been iustified by physical reasomnz but it is clear that it must be possible to derive the theory by a more rigorous mathematical

procedure.

In order to take into account additional effects such as speed and interaction between the sections thè startin oint is ke t me e eneral and therefore it is not assijmed that the ship is sharply pointed. Firstly the steady state prOblem

will be dealt with and: after that the advancing oscillating slender body. 2. Ti STEADY STATE PROBLEM

As derived in chaptçr III the velocity potential consists of two parts. The

first part is the unbounded medium contribution and the second part is the free surface effect:

P3(1,

=

T'

Ref dfÈ(

df/o

( + q + i(if)qsin O +- (L)qcosO

fe

qcos2O

J

qos

dq (5.1)

After changing the order of - and c-integration, a partial integration with

respect to can be carried Out. Omitting higher order terms the result is

(, + )q + f)q sin O + - (tr q cos O

fe

- cos O

dq

31q

O - i

L

n this derivation the bow and stern regions are supposed to be of order s and

therefôre is. of order unity if E is close to the bow or the stérn. If these regions are of order unity or if is situated at a distancé of order unity from the end points the expression (5 2) becomes zero in the hnutrng case

From these facts it must be concluded that the series expansion is not uniformly

convergent in the neigbourhood of

= i

The terms omitted in the above

expression have essentialiy'the form

ab a b a

1 cos u

f

da/sin

- dq

f

_ = O(c In s) (5.2) ob a b a Fq sinu and

ff cs - dq

=

cf

- du

O(s)

These integrals can be neglectéd siñce the first one i of order s in s and the second one is of order s It is difficult to determine the source strength F in (5 2) at bow and stern line The possibility could be considered to choose a constant value for F and to adapt this value to the experimental result

Without attempting a complete solution, some interesting features can be

noted It is clear that the only important free surface disturbance is generated by the bow and stern of the ship Therefore it may be expected that the wave resistance is also caused by the bow and stern effects only For the wave resist-añcé is obtained:

R =

_{(f1f ±ff)}

)F(, )d

r6 Fb

r

r

+ )q + i(rj, f)q sin o ±L (, )q cos.0

ç2

_re'

_{cos O} e

idOl

dq

..j

j 3qcosO1

-

L

The wavé resistance is in the first instance affected only by the source

distribu-tion at the bow and the stern lines From this fact it becomes clear that it must be possible to reduce the wave resistance by adding other singularities in the

bow and stern region, which have the same character as the integrand of (5 2) The strength might be determined by a condition of minimum wave resistance

Following thi procedure the concept of a bulbous bow and stern cOuld be (5.3)

treated in the främe work of slender body theory. Many authors [29], [30], used a dipole to represent the bow or stern bulb. However it is felt that one should be

very careful in applying this procedure for a dipole close to the water surface. Difficulties may be expected due to the linearization of the free surface condi-tion, because it is not a priori clear that a dipole isa correct representation ¿for

a closed body.

3. THE UNSTEADY CASE y 0.25

The Green's fünction, associated with the velocity potential _P2' can be transformed into a slightly different form by separating the poles and in-troducing a new variable:

G2(1, m' i;

, ',

=

00 00 2i

f

dO (

+)q +-(çi)qco50

J

4y cos O - 1J

q -

q -

2) qe

còs{(1 J)q sin O} dq +

- q,(0 + )q + - coo O

2 qdq

2 _qdO

Tfq

_{j e}

cos {q2(1

J)q sin O}\,f

cos O

+

q.(1 +.)q--q1)qos0

e

.cos{q.flqsin.O}.,,1

-j4cos

± G(,'i,

; ) (54)

where the contoursL1

ad

L2are defined by

L1 L2

and the function G has the form: G(1, , , ,

) =

i r qdq

qj(0 + )qi+ -q0q{(r00) cosO + ( f)sin o} q1dO

rJqij

e

Vl_4ycosot

L1 0 1

( qdq

q1(0 +)q -1--qq{(1 ) coo 0(mf)sino}

qd0

' icJ

q

iJ

e

Vi

4rcos O

+

r.jqlj

1 r qdq

i

le

r

vl+4ycosO

/

q3dO

+

O i qdq

q1 + )q - qq {( ) coo 6 (m f)

o} q3dO

+

q e Vi ± 4y cos 0 ¶5.5) L0 O

The quantities q in this chapter are multiplied by a factor compared with those in chapter III and are of order unity. In the formula for the velocity potential

P2

=f dfF(, )G2d

the first three integrals can be integrated with respect to because the remaining

integrals are convergent. The bow and stern region are again assumed to be of order .Neglecting the higher order terms the result is:

r0

f_f}'

rb O

d?f

dO cos 0V41 cos O - i 0O f(q±1

q±2) e

cos{(_f)qsin O}dq +

q( + )q + ! q0()q cosO dO

2i(f

+

-f

)dfq fe

cos {q2(1f)q sin 8}

+

cos 0\/i-4ycos0

r0 rb q0( +)q_q,(s---)qcos6 dO

+

{f_f }F ?3dfq 1fe

cos {q4(1J)q sin 0}

cosO v'i +4 cosO

+

+fdfF(E2Z)G d?

(5.6)

From this expression it can already be concluded that there is a three-dimensi-'anal and forward s eed effect in the velocity potential. This effect is located ip

t e

ow and stern ren and it disappears

the body is sharply pointed.

The integral ¡can be written in the form: o q (

+)q

-q1(E)qcosO 2 qdq e q1dO 1(E1,

=

fdf

q -

e cos{q11fq sin O} d +

O

---6 q(+)q

2 qdq 2 e q dO q

+

-+

43

_0fe

_{cos{q3(1 f)q sin O}d +}

where

In =

7h'

) =fdfF(, )G d

(5.7) An integration with respect to cannot be carried out here since the integrals become divergent at the upper iimitj Therefore another method will be used.

Firstly the order of integration will be changed:

1(,

i,

) =J'df F(, )G d

where D() is the cOntour represented in the sketch below:

Th q)

q

2

_fdfBn

O)e dO (5.8)